11 HOMOGENEOUS SPACES 53
coset x = xH (resp. x = Hx) in G/H (resp. H\G) and each s eG, write
s • i = C&x)H (resp. x - ^ = H(xy)). Then it is clear that G acts on the left
(resp. on the right) on G/H (resp. H\G), and that this action is transitive. The
set G/H (resp. H\G) together with this action of G is called the homogeneous
space of left (resp. right) cosets ofH in G.
We shall work throughout with G/H. Everything we shall do can be
transposed in an obvious way to apply to H\G. Let n denote the canonical
mapping x\-+ xH of G onto G/H.
Clearly G/H is the set of orbits of elements of G for the right action
(A, x) i-» xh (12.10.13) of H on G. If G is a topological group, we can therefore
topologize G/H with the topology defined in (12.10), which is called the
quotient by H of the topology of G. If x0 e G and if x0 = jc0H is its image in
G/H, we obtain a fundamental system of neighborhoods of x0 in G/H by
considering the canonical images in G/H of the neighborhoods V of x0 in G
(i.e., for each V, the set of cosets xH of the elements x e V, or equivalently the
image of V (or of VH) under the canonical mapping n : G -> G/H). Whenever
we consider G/H as a topological space, it will always be this topology that is
meant, unless the contrary is expressly stated.
(12.11.1) The group G acts continuously on G/H.
Let (s0 , x0) be a point of G x (G/H) and let x0 be an element of the
coset x0. Every neighborhood of s0 • x0 is of the form ?r(V), where V is a
neighborhood of ,y0 x0 in G. There exists a neighborhood U of s0 and a
neighborhood W of x0 such that the relations s e U and x e W imply sx e V;
hence the relations s e U and x e 7t(W) imply s - x e 7i(V).
(12.11.2) Let G be a topological group, H a subgroup ofG.
(i) G/H is Hausdorff if and only ifH is closed.
(ii) G/H is discrete if and only ifH is open.
(in) IfH is discrete, then every xeG has a neighborhood V such that the
restriction of n to V is a homeomorphism of V onto the neighborhood n(V) of
n(x) = x in G/H.
(i) If G/H is Hausdorff, then (n(e)} is closed in G/H (12.3.4) and hence
H = n~l(n(e)) is closed in G. Conversely, if H is closed in G, the set of pairs
(x, y) e G x G belonging to the same orbit under the action of H on the right
is the set of (x, y) such that x^lyeH, and is therefore closed, being the
inverse image of H under the continuous mapping (x, y)t-+x~1y. Hence G/H
is Hausdorff (12.10.8).
(ii) If G/H is discrete, the set {n(e)} is open in G/H and therefore
H = n~l(n(e)) is open in G. Conversely, if H is open in G, then so is eachd orbit ofx with respect to M is defined to be the set